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Clustering

Learn about clustering in data analysis, including measuring quality, similarity metrics, and different types of data. Explore partitioning, hierarchical, density-based, and model-based methods.

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Clustering

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  1. Clustering CENG 514

  2. Clustering • Overview • Measuring Quality in Clustering • Similarity Metrics/ Types of Data in Clustering • Basic Categorization of Clustering Methods • Partitioning Methods • Hierarchical Methods • Density-Based Methods • Model-Based Methods • Outlier Analysis 12/20/2019 Clustering 2

  3. What is Clustering? • Cluster: a collection of data objects • Similar to one another within the same cluster • Dissimilar to the objects in other clusters • Clustering/Cluster analysis • Finding similarities between data according to the characteristics found in the data and grouping similar data objects into clusters • Unsupervised learning: no predefined classes 12/20/2019 Clustering 3

  4. Important Issues • Scalability • Ability to deal with different types of attributes • Ability to handle dynamic data • Discovery of clusters with arbitrary shape • Minimal requirements for domain knowledge to determine input parameters • Able to deal with noise and outliers • Insensitive to order of input records • High dimensionality • Incorporation of user-specified constraints • Interpretability and usability 12/20/2019 Clustering 4

  5. Quality of Clustering • A good clustering method will produce high quality clusters with • high intra-class similarity • low inter-class similarity • The quality of a clustering majorly result depends on both the similarity measure used by the method and its implementation 12/20/2019 Clustering 5

  6. Measure the Quality of Clustering • Dissimilarity/Similarity metric: Similarity is expressed in terms of a distance function, typically metric: d(i, j) • The definitions of distance functions are usually very different for interval-scaled, boolean, categorical, ordinal,etc. variables. • Weights should be associated with different variables based on applications and data semantics. • It is hard to define “similar enough” or “good enough” • the answer is typically highly subjective. 12/20/2019 Clustering 6

  7. Data Structures • Data matrix • (obj vs. attr) • Dissimilarity matrix • (obj vs. obj) • distances 12/20/2019 Clustering 7

  8. Type of data in clusteranalysis • Interval-scaled variables • Binary variables • Nominal, ordinal, and ratio variables • Variables of mixed types 12/20/2019 Clustering 8

  9. Interval-valued variables • Normalize data (e.g. z-score normalization) • Calculate the mean absolute deviation: where • Calculate the standardized measurement (z-score) • Using mean absolute deviation is more robust than using standard deviation 12/20/2019 Clustering 9

  10. Similarity and Dissimilarity Between Objects • Distances are normally used to measure the similarity or dissimilarity between two data objects • Some popular ones include: Minkowski distance: where i = (xi1, xi2, …, xip) and j = (xj1, xj2, …, xjp) are two p-dimensional data objects, and q is a positive integer • If q = 1, d is Manhattan distance 12/20/2019 Clustering 10

  11. Similarity and Dissimilarity Between Objects (Cont.) • If q = 2, d is Euclidean distance: • Properties • d(i,j) 0 • d(i,i) = 0 • d(i,j) = d(j,i) • d(i,j)d(i,k) + d(k,j) • Also, one can use weighted distance 12/20/2019 Clustering 11

  12. Similarity and Dissimilarity Between Objects (Cont.) • Example: X1 = (1,2) X2 = (3,5) Euclidean distance(X1, X2) = sqrt(4+9) = 3.61 Manhattan distance(X1, X2) = 2+3 = 5 12/20/2019 Clustering 12

  13. Object j Object i Binary Variables • A contingency table for binary data • Distance measure for symmetric binary variables: • Distance measure for asymmetric binary variables: • Jaccard coefficient (similarity measure for asymmetric binary variables): 13 12/20/2019 Clustering

  14. Dissimilarity between Binary Variables • Example • gender is a symmetric attribute • the remaining attributes are asymmetric binary • let the values Y and P be set to 1, and the value N be set to 0 12/20/2019 Clustering 14

  15. Nominal (Categorical) Variables • A generalization of the binary variable in that it can take more than 2 states, e.g., red, yellow, blue, green • Simple matching • m: # of matches, p: total # of variables 12/20/2019 Clustering 15

  16. Nominal (Categorical) Variables • Example: 12/20/2019 Clustering 16

  17. Nominal (Categorical) Variables • Example: 12/20/2019 Clustering 17

  18. Ordinal Variables • An ordinal variable can be discrete or continuous • Order is important, e.g., rank • Can be treated like interval-scaled • replace xif by their rank • map the range of each variable onto [0, 1] by replacing i-th object in the f-th variable by • compute the dissimilarity using methods for interval-scaled variables 12/20/2019 Clustering 18

  19. Ordinal Variables • Example: Assume that there is an ordering fair <good<excellent 12/20/2019 Clustering 19

  20. Ordinal Variables • Example: Assume that there is an ordering fair <good<excellent 12/20/2019 Clustering 20

  21. Variables of Mixed Types • A database may contain all of these types of variables • symmetric binary, asymmetric binary, nominal, ordinal, interval • One may use a weighted formula to combine their effects • f is binary or nominal: dij(f) = 0 if xif = xjf , or dij(f) = 1 otherwise • f is interval-based: use the normalized distance • f is ordinal • compute ranks rif and • and treat zif as interval-scaled 12/20/2019 Clustering 21

  22. Vector Objects • Vector objects: keywords in documents, gene features in micro-arrays, etc. • Broad applications: information retrieval, biologic taxonomy, etc. • Cosine measure: Given two vectors of attributes, A and B, the cosine similarity is represented using a dot product and magnitude • A variant: Tanimoto coefficient.it yields the Jaccard coefficient in the case of binary attributes 12/20/2019 Clustering 22

  23. Vector Objects Example: Given two vectors: x = (1, 1, 0, 0) y = (0, 1, 1, 0) s(x, y) = (0+1+0+0) / sqrt(2) * sqrt(2) = 0.5 12/20/2019 Clustering 23

  24. Typical Alternatives to Calculate the Distance between Clusters • Single link: smallest distance between an element in one cluster and an element in the other, i.e., dis(Ki, Kj) = min(tip, tjq) • Complete link: largest distance between an element in one cluster and an element in the other, i.e., dis(Ki, Kj) = max(tip, tjq) • Average: avg distance between an element in one cluster and an element in the other, i.e., dis(Ki, Kj) = avg(tip, tjq) • Centroid: distance between the centroids of two clusters, i.e., dis(Ki, Kj) = dis(Ci, Cj) • Medoid: distance between the medoids of two clusters, i.e., dis(Ki, Kj) = dis(Mi, Mj) • Medoid: one chosen, centrally located object in the cluster 12/20/2019 Clustering 24

  25. Centroid, Radius and Diameter of a Cluster (for numerical data sets) • Centroid: the “middle” of a cluster • Radius: square root of average distance from any point of the cluster to its centroid • Diameter: square root of average mean squared distance between all pairs of points in the cluster 12/20/2019 Clustering 25

  26. Major Clustering Approaches • Partitioning approach: • Construct various partitions and then evaluate them by some criterion, e.g., minimizing the sum of squared distances • Typical methods: k-means, k-medoids, CLARANS • Hierarchical approach: • Create a hierarchical decomposition of the set of data (or objects) using some criterion • Typical methods: Diana, Agnes, BIRCH, ROCK, CAMELEON • Density-based approach: • Based on connectivity and density functions • Typical methods: DBSACN, OPTICS, DenClue 12/20/2019 Clustering 26

  27. Partitioning Algorithms: Basic Concept • Partitioning method: Construct a partition of a database D of n objects into a set of k clusters, s.t., min sum of squared distance • Given a k, find a partition of k clusters that optimizes the chosen partitioning criterion • Global optimal: exhaustively enumerate all partitions • Heuristic methods: k-means and k-medoids algorithms • k-means (MacQueen’67): Each cluster is represented by the center of the cluster • k-medoids or PAM (Partition around medoids) (Kaufman & Rousseeuw’87): Each cluster is represented by one of the objects in the cluster 12/20/2019 Clustering 27

  28. The K-Means Clustering Method • Given k, the k-means algorithm is implemented in four steps: • Partition objects into k nonempty subsets • Compute seed points as the centroids of the clusters of the current partition (the centroid is the center, i.e., mean point, of the cluster) • Assign each object to the cluster with the nearest seed point • Go back to Step 2, stop when no more new assignment 12/20/2019 Clustering 28

  29. 10 9 8 7 6 5 4 3 2 1 0 0 1 2 3 4 5 6 7 8 9 10 The K-Means Clustering Method • Example 10 9 8 7 6 5 Update the cluster means Assign each objects to most similar center 4 3 2 1 0 0 1 2 3 4 5 6 7 8 9 10 reassign reassign K=2 Arbitrarily choose K object as initial cluster center Update the cluster means 12/20/2019 Clustering 29

  30. Comments on the K-Means Method • Strength:Relatively efficient: O(tkn), where n is # objects, k is # clusters, and t is # iterations. Normally, k, t << n. • Comparing: PAM: O(k(n-k)2 ), CLARA: O(ks2 + k(n-k)) • Comment: Often terminates at a local optimum. • Weakness • Applicable only when mean is defined, (problem to find the mean for categorical data) • Need to specify k, the number of clusters, in advance • Unable to handle noisy data and outliers • Not suitable to discover clusters with non-convex shapes 12/20/2019 Clustering 30

  31. Variations of the K-Means Method • A few variants of the k-means which differ in • Selection of the initial k means • Dissimilarity calculations • Strategies to calculate cluster means • Handling categorical data: k-modes (Huang’98) • Replacing means of clusters with modes • Using new dissimilarity measures to deal with categorical objects • Using a frequency-based method to update modes of clusters • A mixture of categorical and numerical data: k-prototype method 12/20/2019 Clustering 31

  32. 10 10 9 9 8 8 7 7 6 6 5 5 4 4 3 3 2 2 1 1 0 0 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 Outlier Problem • The k-means algorithm is sensitive to outliers ! • Since an object with an extremely large value may substantially distort the distribution of the data. • K-Medoids: Instead of taking the mean value of the object in a cluster as a reference point, medoids can be used, which is the most centrally located object in a cluster. 12/20/2019 Clustering 32

  33. The K-MedoidsClustering Method • Find representative objects, called medoids, in clusters • PAM (Partitioning Around Medoids, 1987) • starts from an initial set of medoids and iteratively replaces one of the medoids by one of the non-medoids if it improves the total distance of the resulting clustering • PAM works effectively for small data sets, but does not scale well for large data sets • CLARA (Kaufmann & Rousseeuw, 1990) • CLARANS (Ng & Han, 1994): Randomized sampling 12/20/2019 Clustering 33

  34. 10 10 9 9 8 8 7 7 6 6 5 5 4 4 3 3 2 2 1 1 0 0 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 A Typical K-Medoids Algorithm (PAM) Total Cost = 20 10 9 8 Arbitrary choose k object as initial medoids Assign each remaining object to nearest medoids 7 6 5 4 3 2 1 0 0 1 2 3 4 5 6 7 8 9 10 K=2 Randomly select a nonmedoid object,Oramdom Total Cost = 26 Do loop Until no change Compute total cost of swapping Swapping O and Orandom If quality is improved. 12/20/2019 Clustering 34

  35. What Is the Problem with PAM? • Pam is more robust than k-means in the presence of noise and outliers because a medoid is less influenced by outliers or other extreme values than a mean • Pam works efficiently for small data sets but does not scale well for large data sets. • O(k(n-k)2 ) for each iteration where n is # of data,k is # of clusters • Sampling based method, CLARA(Clustering LARge Applications) 12/20/2019 Clustering 35

  36. Step 1 Step 2 Step 3 Step 4 Step 0 agglomerative (AGNES) a a b b a b c d e c c d e d d e e divisive (DIANA) Step 3 Step 2 Step 1 Step 0 Step 4 Hierarchical Clustering • Use distance matrix as clustering criteria. This method does not require the number of clusters k as an input, but needs a termination condition 12/20/2019 Clustering 36

  37. AGNES (Agglomerative Nesting) • Introduced in Kaufmann and Rousseeuw (1990) • Use the Single-Link method and the dissimilarity matrix. • Merge nodes that have the least dissimilarity • Eventually all nodes belong to the same cluster 12/20/2019 Clustering 37

  38. Dendrogram: Shows How the Clusters are Merged Decompose data objects into a several levels of nested partitioning (tree of clusters), called a dendrogram. A clustering of the data objects is obtained by cutting the dendrogram at the desired level, then each connected component forms a cluster. 12/20/2019 Clustering 38

  39. DIANA (Divisive Analysis) • Introduced in Kaufmann and Rousseeuw (1990) • Inverse order of AGNES • Eventually each node forms a cluster on its own 12/20/2019 Clustering 39

  40. Well-known Hierarchical Clustering Methods • Major weakness of agglomerative clustering methods • do not scale well: time complexity of at least O(n2), where n is the number of total objects • can never undo what was done previously • Integration of hierarchical with distance-based clustering • BIRCH (1996): uses CF-tree and incrementally adjusts the quality of sub-clusters • ROCK (1999): clustering categorical data by neighbor and link analysis • CHAMELEON (1999): hierarchical clustering using dynamic modeling 12/20/2019 Clustering 40

  41. BIRCH (1996) • Birch: Balanced Iterative Reducing and Clustering using Hierarchies (Zhang, Ramakrishnan & Livny, SIGMOD’96) • Incrementally construct a CF (Clustering Feature) tree, a hierarchical data structure for multiphase clustering • Phase 1: scan DB to build an initial in-memory CF tree (a multi-level compression of the data that tries to preserve the inherent clustering structure of the data) • Phase 2: use an arbitrary clustering algorithm to cluster the leaf nodes of the CF-tree • Scales linearly: finds a good clustering with a single scan and improves the quality with a few additional scans • Weakness: handles only numeric data, and sensitive to the order of the data record. 12/20/2019 Clustering 41

  42. Clustering Feature:CF = (N, LS, SS) N: Number of data points LS: Ni=1=Xi SS: Ni=1=Xi2 Clustering Feature Vector in BIRCH CF = (5, (16,30),(54,190)) (3,4) (2,6) (4,5) (4,7) (3,8) 12/20/2019 Clustering 42

  43. CF-Tree in BIRCH • Clustering feature: • summary of the statistics for a given subcluster: the 0-th, 1st and 2nd moments of the subcluster from the statistical point of view. • registers crucial measurements for computing cluster and utilizes storage efficiently • A CF tree is a height-balanced tree that stores the clustering features for a hierarchical clustering • A nonleaf node in a tree has descendants or “children” • The nonleaf nodes store sums of the CFs of their children • A CF tree has two parameters • Branching factor: specify the maximum number of children. • threshold: max diameter of sub-clusters stored at the leaf nodes 12/20/2019 Clustering 43

  44. CF1 CF2 CF3 CF6 child1 child2 child3 child6 The CF Tree Structure Root B = 7 L = 6 Non-leaf node CF1 CF2 CF3 CF5 child1 child2 child3 child5 Leaf node Leaf node prev CF1 CF2 CF6 next prev CF1 CF2 CF4 next 12/20/2019 Clustering 44

  45. Density-Based Clustering Methods • Clustering based on density (local cluster criterion), such as density-connected points • Major features: • Discover clusters of arbitrary shape • Handle noise • One scan • Need density parameters as termination condition • Well-known examples: • DBSCAN: Ester, et al. (KDD’96) • OPTICS: Ankerst, et al (SIGMOD’99). • DENCLUE: Hinneburg & D. Keim (KDD’98) • CLIQUE: Agrawal, et al. (SIGMOD’98) (more grid-based) 12/20/2019 Clustering 45

  46. p MinPts = 5 Eps = 1 cm q Density-Based Clustering: Basic Concepts • Two parameters: • Eps: Maximum radius of the neighbourhood • MinPts: Minimum number of points in an Eps-neighbourhood of that point • NEps(p): {q belongs to D | dist(p,q) <= Eps} • Directly density-reachable: A point p is directly density-reachable from a point q w.r.t. Eps, MinPts if • p belongs to NEps(q) • core point condition: |NEps (q)| >= MinPts 12/20/2019 Clustering 46

  47. p q o Density-Reachable and Density-Connected • Density-reachable: • A point p is density-reachable from a point q w.r.t. Eps, MinPts if there is a chain of points p1, …, pn, p1 = q, pn = p such that pi+1 is directly density-reachable from pi • Density-connected • A point p is density-connected to a point q w.r.t. Eps, MinPts if there is a point o such that both, p and q are density-reachable from o w.r.t. Eps and MinPts p p1 q 12/20/2019 Clustering 47

  48. Outlier Border Eps = 1cm MinPts = 5 Core DBSCAN: Density Based Spatial Clustering of Applications with Noise • Relies on a density-based notion of cluster: A cluster is defined as a maximal set of density-connected points • Discovers clusters of arbitrary shape in spatial databases with noise 12/20/2019 Clustering 48

  49. DBSCAN: The Algorithm • Arbitrary select a point p • Retrieve all points density-reachable from p w.r.t. Eps and MinPts. • If p is a core point, a cluster is formed. • If p is a border point, no points are density-reachable from p and DBSCAN visits the next point of the database. • Continue the process until all of the points have been processed. 12/20/2019 Clustering 49

  50. DBSCAN: Sensitive to Parameters 12/20/2019 Clustering 50

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